2.10 Recap and reflection

In this chapter we introduced many important concepts and results which form the backbone of much of the material encountered throughout a mathematics degree. The $\varepsilon$-$N$ definition of a limit made precise the intuitive idea of a sequence whose terms get ‘closer and closer’ to some fixed value. We first familiarised ourselves with this definition by working with explicit examples. We then proceeded to develop a general theory of limits, based on this definition.

Using abstract $\varepsilon$-$N$ arguments, we proved many important results about limits:

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  The limit laws and the squeeze theorem are used to compare complicated sequences with simpler sequences which are easier to understand.

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  The boundedness and subsequence test provide useful criteria for divergence of sequences.

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  The monotone convergence theorem and Cauchy criterion allow one to prove limits exist without having to know the explicit value of the limit.

In the next chapter we shall study series, which are a special kind of sequence formed by repeated addition. We shall see further applications of all the above results in this context, and the monotone convergence theorem and Cauchy criterion will play a particularly important role.